Liouvillian Exceptional Points in Quantum Systems

• Liouvillian Exceptional Points (LEP) are non-Hermitian degeneracies in open quantum systems where eigenvalues and eigenmodes coalesce into Jordan block structures.
• LEPs induce anomalous dynamics such as non-exponential relaxation and super-Lorentzian spectral features, making them pivotal for advanced spectroscopic diagnostics.
• Experimental studies in collective spin systems leverage LEPs to identify spectral phase transitions and optimize model validation using criteria like the Bayesian Information Criterion.

A Liouvillian Exceptional Point (LEP) is a non-Hermitian degeneracy in the spectrum of the Lindblad superoperator describing open quantum systems, where two or more eigenvalues and the corresponding eigenmodes coalesce, resulting in a non-diagonalizable Liouvillian with Jordan block structure. The presence of LEPs induces anomalous dynamics including non-exponential relaxation, higher-order poles in response functions, and state-dependent spectroscopic signatures. Recent advances have demonstrated both their fundamental role in quantum many-body dynamics and their experimental accessibility, notably through spectroscopic diagnostics in collective spin systems (Molina, 1 Feb 2026).

1. Definition and Structure of Liouvillian Exceptional Points

Let $\mathcal{L}$ denote the Lindblad (or more general GKSL) generator for Markovian open quantum dynamics: $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$ where $H$ is the system Hamiltonian, $\{L_j\}$ are jump operators, and $\gamma_j$ are the corresponding dissipation rates.

In the Liouville space (vectorized operator space), $\mathcal{L}$ is a (generally non-Hermitian) matrix. Its spectral decomposition is

$$\mathcal{L}|\rho\rangle\rangle = \sum_\mu \lambda_\mu\,\Pi_\mu\,|\rho\rangle\rangle,$$

with $\{\lambda_\mu\}$ the eigenvalues and $\{\Pi_\mu\}$ spectral projectors. When $\mathcal{L}$ is non-diagonalizable, the projectors generalize to Jordan projectors.

A second-order LEP occurs at some parameter value where two eigenvalues $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$0 and associated eigenmatrices coalesce: $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$1 yielding a $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$2 Jordan block. This generalizes to LEPs of order $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$3, corresponding to Jordan blocks of size $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$4.

2. Spectral Consequences and Liouvillian Resolvent

The Liouvillian resolvent, crucial for frequency-domain dynamics, is defined as

$$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$5

Around a non-degenerate eigenvalue $$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$6,

$$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$7

but at an order-$$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$8 LEP,

$$\dot\rho = \mathcal{L}[\rho] = -i[H,\rho] + \sum_j \gamma_j (L_j\,\rho\,L_j^\dagger - \tfrac{1}{2}\{L_j^\dagger L_j, \rho\}),$$9

introducing higher-order (double, triple, etc.) poles (Molina, 1 Feb 2026).

This leads, for example, to super-Lorentzian features in emission spectra: whereas a standard Lorentzian arises from a simple pole, a double pole from a Jordan block gives rise to a "super-Lorentzian". In the collective spin model, the emission spectrum $H$0 takes the form

$H$1

The second term directly signals the presence of a defective Liouvillian mode and hence an LEP.

3. Exceptional Spectral Phase in Many-Body Collective Spin Systems

In the model analyzed in (Molina, 1 Feb 2026), a Dicke collective spin ($H$2) coupled to a polarized Markov bath via three jump operators ($H$3, $H$4) demonstrates an extensive region ("exceptional spectral phase", ESP) with proliferating second-order LEPs in the $H$5 limit and for high bath polarization. A spectral phase transition exists, across a critical polarization $H$6, beyond which large sectors of the Liouvillian spectrum become defective.

This ESP is characterized by a macroscopic number of defective modes, leading to the dominance of non-diagonalizable dynamics throughout the Liouvillian spectrum, which manifests as super-Lorentzian (higher-order) line shapes in the system's fluorescence and spin-noise spectra.

4. Practical Spectroscopic Diagnostics and State Dependence

In frequency-resolved experiments, differentiating Lorentzian from super-Lorentzian line shapes provides a direct probe of LEPs. The weight of the super-Lorentzian component is quantified by the parameter

$H$7

with $H$8 indicating a genuine second-order pole (defective mode contribution).

However, the detectability of LEP fingerprints is highly state-dependent. Steady-state emission can suppress overlap with Jordan subspaces (e.g., in the highly-polarized regime of $H$9), yielding an apparently pure Lorentzian even when the underlying spectrum is defective. By contrast, initial states that sample all symmetry sectors (e.g., infinite-temperature or random initializations) fully reveal the LEP-induced super-Lorentzian features (Molina, 1 Feb 2026).

To maximize detection:

• Prepare initial states populating all symmetry sectors.
• Fit experimental spectra to both Lorentzian-only (Model A) and Lorentzian+super-Lorentzian (Model B) forms, comparing via Bayesian or Akaike Information Criteria; significant negative $\{L_j\}$0BIC indicates an LEP contribution.

5. Symmetry Sector Selection and the Role of Weak Symmetries

The Liouvillian of the collective-spin model commutes with the generator $\{L_j\}$1, decomposing Liouville space into sectors labeled by magnetization $\{L_j\}$2. The physical emission spectrum for observables like $\{L_j\}$3 only probes the $\{L_j\}$4 sector.

This sector selection enables or suppresses the manifestation of defective modes in measurement outcomes, depending on the alignment between the emission process and the Jordan subspace. Thus, not all defective modes present in the full Liouvillian contribute to physically detected quantities, which rigorously constrains both experimental and theoretical interpretation (Molina, 1 Feb 2026).

6. Broader Implications and Experimental Relevance

Identification of LEPs and ESPs provides robust dynamical and spectroscopic markers—a breakdown of strictly exponential relaxation, critical slowing down, and emergence of higher-order poles—for detecting and classifying non-Hermitian degeneracies in many-body open quantum systems.

These findings have implications for:

• Quantum many-body systems exhibiting nontrivial Liouvillian topology and spectral phase transitions.
• Cavity QED ensembles and platforms where spectrally resolved two-point correlation functions can be measured.
• Optimization and performance characterization of quantum thermal machines, where relaxation times are set by the Liouvillian gap and critical damping occurs at LEPs.

Furthermore, the effect of initial state preparation and sector selectivity accentuates the necessity of tailored protocols for unambiguous experimental observation of many-body LEPs (Molina, 1 Feb 2026).